Compound Interest
Compound interest is the process where money earns interest on both the original principal and previously accumulated interest. A $10,000 investment at 7% annual compound interest grows to approximately $38,697 after 20 years, demonstrating how returns accelerate over time.
Why it matters
Compound interest drives long-term wealth building through savings accounts, investment portfolios, and retirement funds. The formula A = P(1 + r)n reveals why starting early matters significantly: $5,000 invested at age 25 earning 6% annually becomes $91,000 by age 65, while the same investment starting at age 35 only reaches $51,000. Financial institutions use compound interest calculations for mortgages, credit cards, and business loans. Index funds averaging 7% returns can double an investment approximately every 10 years through compounding. The concept appears in CCSS.HSF.IF.C.8b, connecting exponential functions to real financial applications. Understanding compound interest helps evaluate retirement planning, college savings plans, and debt management strategies where interest compounds monthly or quarterly.
How to solve compound interest
Compound Interest
- Compound interest earns interest on both the original principal AND on previously earned interest — that's why the curve bends upward over time.
- Annual: A = P(1 + r)n, where P is the principal, r the rate as a decimal, n the number of years.
- Monthly compounding: A = P(1 + r/12)12n.
- With monthly contributions PMT: future value = PMT × [((1 + i)n − 1) / i], where i = r/12 and n is the number of months.
- Index funds and savings accounts both rely on this — small early differences in rate, time, or starting age compound to outsized differences at the end.
Example: $10,000 at 7% for 20 years: A = 10000 · 1.07²⁰ ≈ $38,697.
Worked examples
You deposit $5,000.00 in a savings account paying 3% interest per year. How much is in the account after one year?
Answer: 5150
- Calculate the interest → 5000 × 3/100 = 150 — Interest = principal × rate. $5,000.00 × 3% = $150.00.
- Add the interest to the principal → 5000 + 150 = 5150 — After one year: $5,000.00 + $150.00 = $5,150.00.
You invest $15,000.00 at 6% compound interest per year. What is the value after 2 years?
Answer: 16854
- Use the compound interest formula → A = P(1 + r)^n — P is the principal, r the rate as a decimal, and n the number of years.
- Plug in the values → A = 15000 × (1 + 0.06)^2 — P = $15,000.00, r = 0.06, n = 2.
- Compute the growth factor → (1 + 0.06)^2 = 1.1236 — Raise 1 + r to the power n.
- Multiply by the principal → A ≈ $16,854.00 — $15,000.00 × 1.1236 ≈ $16,854.00 after rounding.
You invest $50,000.00 in an index fund returning 6% per year. What is the value after 7 years, assuming returns are reinvested?
Answer: 75182
- Apply A = P(1 + r)^n → A = 50000(1 + 0.06)^7 — Reinvested returns compound — the formula treats each year's gain as next year's principal.
- Compute the result → A ≈ $75,182.00 — After 7 years, $50,000.00 grows to approximately $75,182.00 — a gain of $25,182.00.
Common mistakes
- Confusing simple and compound interest calculations leads to writing $1,000 at 5% for 3 years as $1,150 instead of the correct compound amount of $1,157.63.
- Forgetting to convert percentage rates to decimals produces errors like calculating $5,000 at 6% as 5000(1.06)^2 = $5,618 instead of using 0.06 for $5,618.
- Using the wrong time period in the exponent causes mistakes such as calculating monthly compounding for 2 years as (1 + 0.05/12)^2 instead of (1 + 0.05/12)^24.
- Mixing up principal and final amount in word problems results in treating the $15,000 final value as the starting principal rather than the ending amount.